WIND TURBINE AIRFOIL OPTIMIZATION PARTICLE SWARM METHOD MAKOTO ENDO. for the degree of Master of Science

Transcription

1 WIND TURBINE AIRFOIL OPTIMIZATION BY PARTILE SWARM METHOD by MAKOTO ENDO Submitted in partial fulfillment of the requirements for the degree of Master of Science Thesis Advisor: Dr. James S. T ien and Dr. Meng-Sing Liou Department of Mechanical and Aerospace Engineering ASE WESTERN RESERVE UNIVERSITY January,

2 ASE WESTERN RESERVE UNIVERSITY SHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of Makoto Endo Master of Science candidate for the degree *. James S. T'ien (signed) (chair of the committee) J. Iwan D. Alexander Meng-Sing Liou Sep. 8, (date) *We also certify that written approval has been obtained for any proprietary material contained therein.

3 Table of ontents Table of ontents List of Tables 5 List of Figures 6 Acknowledgements Nomenclature Abstract 5 hapter : Introduction 6. Optimization and wind turbine 6. Literature review 7.. Optimization methods 7.. Flow field calculation around wind turbine 8..3 Blade parameterization.3 Purpose and scope of this study hapter : Mathematical Model. Particle swarm optimization. Objective function 3.. Aerodynamics of wind turbine 3... Power available from wind 3... Axial moment theory Power created by a blade element 8

4 ...4 Optimization condition for wind turbine airfoil 9.. PARSE airfoil 3... The original PARSE airfoil 3... Modified PARSE airfoil Thickness calculation for PARSE airfoil Flow field calculation RNG k- model Near wall treatment Discretization Grid generation 39.3 Wind turbine airfoil shape optimization 4.3. Flow condition for optimization 4.3. Geometrical parameters for optimization Objective functions for optimization oncept of domination Accuracy check criteria for optimization Unphysical airfoil Airfoil that does not give an accurate solution Maximum thickness of airfoil exceeds its limit Parallel computing by MPI (Message Passing Interface) 45 hapter3: omputed Results Validation of components A test of two objective PSO by Hu and Eberhart 69 3

5 3.. Flow field calculation around S89 airfoil Method of representing S89 airfoil Flow field calculation around S89 airfoil (Euler) Flow field calculation around S89 airfoil (RNG k-) Grid sensitivity OL grid (Outlet length times) IL grid (Inlet length times) D grid (grid density times) B grid (different Boundary ondition) Wind turbine airfoil shape optimization % thick condition % thick condition omparison of % and % thick condition 78 hapter 4: Summary and onclusion 39 Future Work 4 References 4 4

6 List of Tables Table -: l and d comparison for different r le _ up and le low r _ 47 Table 3-: Original S89 airfoil coordinates by Somers 8 Table 3-: Interpolated S89 airfoil coordinates using GAMBIT 8 Table 3-3: Thickness of nondominated solutions in "%thick condition" 8 5

7 List of Figures Figure -: How particle founds its local best 48 Figure -: Air coming into a wind turbine 49 Figure -3: Figure -4: Axial stream tube model Air flow vertically coming into the rotating surface 5 5 Figure -5: Relevant velocity for blade section 5 Figure -6: hord length distribution along the radius direction 53 Figure -7: PARSE airfoil geometry defined by basic parameters 54 Figure -8: Modified PARSE airfoil geometry defined by basic parameters 55 Figure -9: Shape of the three airfoils (case A, B and ) 56 Figure -: Objective function values of the three airfoils (case A, B and ) 57 Figure -: Sketch of camber line 58 Figure -: How thickness is calculated for PARSE airfoil 59 Figure -3: Domain size 6 Figure -4: Grid near the airfoil 6 Figure -5: Boundary conditions 6 Figure -6: Flowchart for wind turbine airfoil shape optimization 63 Figure -7: oncept of domination 64 Figure -8: Example of unphysical airfoil shape 65 Figure -9: Example of how Figure -: Example of how Fx value converges 66 F y value converges 67 6

8 Figure -: Example of unconverged solution 68 Figure 3-: Sketch of the antilever design problem 83 Figure 3-: Particle movement along the increasing number of iteration 84 Figure 3-3: loser look of the Pareto front at iteration 85 Figure 3-4: Figure 3-5: omparison of interpolation by IEM FD and GAMBIT (far view) omparison of interpolation by IEM FD and GAMBIT (close view) Figure 3-6: Figure 3-7: Figure 3-8: P distribution by Euler calculation (angle of attack of [deg]) 88 P distribution by Euler calculation (angle of attack of. [deg]) 89 P distribution by Euler calculation (angle of attack of 5.3 [deg]) 9 Figure 3-9: How F Z not converging at angle of attack of 9. [deg] 9 Figure 3-: Figure 3-: Figure 3-: Figure 3-3: Figure 3-4: P distribution by RNG k-ε calculation (angle of attack of [deg]) P distribution by RNG k-ε calculation (angle of attack of. [deg]) P distribution by RNG k-ε calculation (angle of attack of 5.3 [deg]) P distribution by RNG k-ε calculation (angle of attack of 9. [deg]) P distribution by RNG k-ε calculation (angle of attack of 4.4 [deg]) Figure 3-5: Figure 3-6: l comparison between RNG k-epsilon calculation and experiment 97 d comparison between RNG k-epsilon calculation and experiment 98 Figure 3-7: Grid comparison between "OL" grid and "base grid" 99 7

9 Figure 3-8: Figure 3-9: l comparison between "OL grid" and "base grid" d comparison between "OL grid" and "base grid" Figure 3-: Grid comparison between "IL" grid and "base grid" Figure 3-: Figure 3-: l comparison between "IL grid" and "base grid" 3 d comparison between "IL grid" and "base grid" 4 Figure 3-3: Grid comparison between "D" grid and "base grid" 5 Figure 3-4: Figure 3-5: l comparison between "D grid" and "base grid" 6 d comparison between "D grid" and "base grid" 7 Figure 3-6: How the boundary condition is changed in the B grid 8 Figure 3-7: Figure 3-8: l comparison between "B grid" and "base grid" 9 d comparison between "B grid" and "base grid" Figure 3-9: Function value occurred during optimization and computed function value of S89 airfoil. ( % thick condition) Figure 3-3: Non-dominated function value and computed function value of S89 airfoil. ( % thick condition) Figure 3-3: How non-dominated solutions are named ( % thick condition) 3 Figure 3-3: Shape and performance of - airfoil 4 Figure 3-33: Shape and performance of - airfoil 5 Figure 3-34: Shape and performance of -3 airfoil 6 Figure 3-35: Shape and performance of -4 airfoil 7 Figure 3-36: Shape and performance of -5 airfoil 8 Figure 3-37: Shape and performance of -6 airfoil 9 8

10 Figure 3-38: Shape and performance of -7 airfoil Figure 3-39: Shape and performance of -8 airfoil Figure 3-4: Shape and performance of -9 airfoil Figure 3-4: Shape and performance of - airfoil 3 Figure 3-4: Function value occurred during optimization and computed function value of S89 airfoil. ( % thick condition) Figure 3-43: Non-dominated function value and computed function value of S89 airfoil. ( % thick condition) Figure 3-44: Non-dominated function value that dominates S89 s function value and, computed function value of S89 airfoil ( % thick condition) Figure 3-45: How non-dominated solutions that dominates S89 s function value are named ( % thick condition) Figure 3-46: Shape and performance of - airfoil 8 Figure 3-47: Shape and performance of - airfoil 9 Figure 3-48: Shape and performance of -3 airfoil 3 Figure 3-49: Shape and performance of -4 airfoil 3 Figure 3-5: Shape and performance of -5 airfoil 3 Figure 3-5: Shape and performance of -6 airfoil 33 Figure 3-5: Shape and performance of -7 airfoil 34 Figure 3-53: Shape and performance of -8 airfoil 35 Figure 3-54: Shape and performance of -9 airfoil 36 Figure 3-55: Shape and performance of - airfoil 37 Figure 3-56: omparison of non-dominated solutions obtained by % thick condition and % thick condition 38 9

11 Acknowledgements I wish to express my gratitude to my advisor, Prof. James S. T ien for his invaluable guidance and encouragement throughout this work. I would also like to thank Dr. Meng-Sing Liou at NASA Glenn Research enter whose meticulous comments were an enormous help to me. I am grateful as well to Prof.Alexander for his suggestions and support. This research began as a summer project at Multidisciplinary Analysis, Design and Optimization Institute Student Residencies at NASA Glenn Research enter. I wish to thank MADO institute for their generous financial assistance and giving me the opportunity to interact with many inspiring researchers. Among them, I especially want to thank Dr. Hyoungjin Kim, Dr. Byung Joon Lee and Dr. Takayasu Kumano for their helpful comments. Finally, I would like to thank my fellow summer intern Brendan Tracy from Stanford University and my ase Western Reserve University colleagues Dr. Sheng-Yen Hsu, Ya-Ting Tseng and Michael Johnston for frequent, stimulating and helpful discussions.

12 Nomenclature A Area of section - (see Fig.-3) A Area of section - (see Fig.-3) A T a B l Rotating surface area of the rotor (see Fig.-3) Axial induction factor Number of blades hord length Lift coefficient d Drag coefficient p Pressure coefficient Power coefficient power D Drag d Diameter of the cantilever (see Fig. 3-) dq dr E F F x Differential torque created by the blade element Width of a blade element in the radius direction (see Fig.-5) Young's modulus Thrust force experienced by the rotor Force in the direction of chord line F y Force vertical to the chord line k Turbulence kinetic energy

13 L Lift L Best position of particle found so far by each swarm member selected best from the "closest" two swarm members (see section.) l Length of the cantilever (see Fig. 3-) N P Total number of particles used for the optimization Pressure P Force applied at the end of the cantilever (only used in section 3..) (see Fig. 3-) P Theoretical power available in a wind stream air P Best position of particle found so far by each of the particles best (see section.) P D Pressure at the downstream of the rotor (see Fig.-3) P T Power developed by the wind turbine P U R Re r r le Pressure at the upstream of the rotor (see Fig.-3) Total length of the rotor in the radius direction (center of rotation to tip) Reynolds number Distance from the center of rotation to the blade element (see Fig.-5) PARSE parameter that describes the leading edge radius r _ Modified PARSE parameter that describes the r le for the upper curve le up (see Fig. -8) r _ Modified PARSE parameter that describes the r le for the lower curve le low (see Fig. -8)

14 S y Allowable strength U u v V Inlet velocity Flow velocity in x-direction Flow velocity in y-direction Wind velocity at section - (see Fig.-3) V Wind velocity at section - (see Fig.-3) V Relative flow velocity with respect to airfoil (see Fig.-5) rel V T Velocity at the turbine section (see Fig.-3) x lo PARSE parameter that describes the x location that gives the lowest z value (see Fig. -7) x up PARSE parameter that describes the x location that gives the highest z value (see Fig. -7) z PARSE parameter that describes the lowest z value (see Fig. -7) lo z PARSE parameter that describes the z location at x= (see Fig. -7) TE z PARSE parameter that describes the highest z value (see Fig. -7) up z PARSE parameter that describes the second derivative of the lower xxlo curve at x lo (see Fig. -7) z PARSE parameter that describes the second derivative of the upper xxup curve at x up (see Fig. -7) Angle of attack PARSE parameter that describes the direction of the trailing edge TE 3

15 (see Fig. -7) PARSE parameter that describes the angle of the trailing edge TE (see Fig. -7) E kinetic energy of a stream of air coming into a wind turbine z TE PARSE parameter that describes the thickness at the trailing edge (see Fig. -7) Deflection Deflection limit max a Dissipation rate of turbulence kinetic energy Tip speed ratio Dynamic viscosity of the air Angle between Vrel and r (see Fig.-5) Density Density of air Rotating speed of the wind turbine 4

16 Wind Turbine Airfoil Optimization by Particle Swarm Method Abstract by MAKOTO ENDO Two-dimensional shape of a wind turbine blade was optimized by means of Particle Swarm Optimization. By following blade element theory, lift coefficient l and drag coefficient d were used as objective functions. In order to compute the objective functions, flow field around airfoils were calculated by Re-Normalization Group (RNG) k-ε model. Shapes of airfoils were defined by modified PARSE method with parameters. Two optimization cases were conducted with maximum thickness limited to % and % of the chord length respectively. In both cases, Reynolds number was set 6 at., which is the design condition of S89 airfoil. S89 airfoil is a well known airfoil used in wind turbines and many experimental data are available. The angle of attack for the optimization was set at 5.3 deg., the mount angle of S89. Non-dominated solutions obtained by this research were compared with the performance of S89 at several angles of attack. The results of optimization showed that ) there is a strong influence of maximum thickness of airfoil to its performance, ) non-dominated solutions constitute a gradual relationship which implies that there are many airfoil shapes that could be considered as an optimum. The resulting shape along this Pareto front showed higher performance than the existing blade section (i.e. NREL S89) in certain conditions. 5

17 hapter Introduction. Optimization and wind turbine Most engineering problems are multi-objective in nature. Design objectives are often conflicting and may have tradeoffs among them. For instance, when we design an airfoil, we may want the lift, drag, stall angle, etc., to be optimized. Although the current computational fluid dynamics codes are capable of solving these values for a given shape, in most cases, the trade off relation between each shape is estimated by the researcher. The recent increase of computational power has enabled us to run many calculations in parallel; though ironically, has made it harder for researchers to estimate the relations between cases. A simple, robust optimization scheme that is suitable for multi-objective problem in parallel computation is therefore highly desirable. Application of optimization techniques to wind turbines has additional incentives compared to some other fields. In airplanes, the relative wind power that reaches the wing is mainly due to the engine, the performance can be determined only by the airplane itself (although gust, icing, and sand erosion are still a problem). For wind turbines, the situation is much more passive since wind turbines are built to bring out energy from wind. The performance of the wind turbine highly depends on the velocity, angle, and steadiness of the wind. What makes the problem much more complicated and interesting is that this performance of wind depends not only on geographical factors, but also on the interaction with nearby wind turbines. By this characteristic, the "optimal wind turbine" cannot be decided in a single way. A method that calculates the desired shape from the performance of its surroundings is needed. Wind turbines are a suitable target for optimization. 6

18 . Literature review.. Optimization methods Multi-objective optimization can be divided into Genetic Algorithm (GA) and Non-Genetic Algorithm. Features of Genetic Algorithm are: Mixed variable types can be handled ontinuous and discrete function can be treated together Implicit parallelism (since it is population based) Probabilistic operators (less chance of getting stuck) GA is an especially a promising approach for aerospace applications because of the following reasons: Since FD calculations tends to require time, there is a great advantage in doing parallel computing. Engineering parameters, like gear ratio, will take a discrete value; although, geometrical parameters are likely to be continuous. In addition, the problem may contain mixed variables. For instance, in 4, Oyama, Liou and Obayashi optimized the transonic axial flow blade shape and were able to reduce the entropy production by more than 9% (compared with the NASA rotor67); while, still satisfying constraints on the mass flow rate and the pressure ratio []. The optimization scheme used in this research is a kind of GA called Particle Swarm Optimization (PSO). It is a relatively new method that was originally introduced 7

19 by Eberhart and Kennedy []. For this research, PSO scheme by Hu and Eberhart [3] was used. Although detailed information of this method will be explained later, the general characteristic of PSO can be said to be that it is a GA that is tuned more towards converging speed than robustness. Fast convergence is especially important for an optimization process involving computation-intensive FD. From experimental back ground, we know that Pareto-front will not be a complicated shape... Flow field calculation around wind turbine In the process of optimization, getting an accurate value of the objective function, or at least the right tendency of the objective function is important. In order to validate the process of getting the properties of the blade shape, I have chosen the NREL S89 airfoil as the reference airfoil to compare with the optimized results. The S89 airfoil was designed for horizontal axis wind turbine [5], and many wind-tunnel experiments were conducted using this airfoil at Delft University of Technology, Ohio State University and olorado State University [4]. The S89 airfoil has been the subject of several numerical analyses. Guerri et al. gave a great review of this subject [7]. The performance of different calculation methods can be summarized as follows: Euler simulation (inviscid calculation): Wolfe and Ochs [8] performed D steady FD simulations of S89 characteristics using a commercial code FDAE. In their work, comparison was done between inviscid Euler calculation, viscous laminar/turbulent calculation, and fully turbulent calculation. For low angles of attack when the flow is attached to the airfoil, the Euler calculation gave a reasonable pressure coefficient ( ) distribution. This result is p 8

20 reproduced in section 3... Although lift coefficient l matched well with experimental data, drag coefficient drag. Reynolds Averaged Numerical Simulation (RANS): d was underestimated due to neglecting the viscous Incompressible Re Normalization Group (RNG) k-ε and Shear Stress Transport (SST) k-ω model are compared by Guerri et al [7]. Their results showed that: () at low angle of attack before separation, both turbulence models give good accuracy for l and d ; () at high angle of attack after separation, the SST k-ω model shows better results compared to the RNG k-ε model, although they are both somewhat different compared to experimental data; (3) the value of l and fluctuate, while the value obtained by the RNG k-ε model is stable. Large Eddy Simulation (LES): d obtained by the SST k-ω model will According to Mellen et al. [9]: "When resolution requirements are specified, LES is able to produce the correct overall flow behavior. Details are provided on the structure of the flow and its time-dependant behavior that are not available from RANS calculations. However, meeting resolution requirements lead to calculations that are extremely expensive and currently not suitable for routine use ". Detached Eddy Simulation (DES): DES is a mixture of RANS and LES strategy. Johansen et. al. [6] compared the result by (i) the Shear Stress Transport (SST) k/ω model of Menter and (ii) a DES model based on the SST k/ω model of Menter against a flow around a wind turbine blade based on S89 airfoil (fixed angle of attack and pitching along the blade axis). It was shown 9

21 that the DES computations did not improve the predicted blade characteristics compared to the SST k/ω model. The optimization in this thesis was done by RNG k-εmodel. Because LES takes too much computational time and Euler calculation will significantly underestimate the d value. We compared the result by k-εmodel against experimental data taken in wind tunnel [5] and the general characteristics are matched. This will be discussed in Section Blade parameterization To optimize the shape of wind turbine blade section, the way to define the shape has to be considered. The requirements for a good parameterization are: There should be robustness in expressing the blade shape. If not, there will be a possibility that the actual optimum shape cannot be described. The number of parameters to describe the blade shape must be small as enough from the point of computational efficiency. Obviously, and are conflicting matters and the best way to parameterize the shape has yet to be decided. The most common way to parameterize a blade shape is to use β-spline curve. For instance, Oyama et al. used 4 parameters and β-spline function to describe transonic axial-flow blade shape [].

22 The parameterization method used in this research is called PARSE, which was introduced by Sobieczky []. One of the advantages of the PARSE method is that each parameter corresponds to an aerodynamically meaningful value (leading edge radius, thickness, etc.). Another strength is that only parameters are needed to describe a section shape. To our knowledge, PARSE has never been utilized for wind turbines. To determine the effectiveness of this method is one of the objectives of this research. Through this research, we noticed that there should be some modification in PARSE to find a good shape for wind turbine. This will be discussed in Section....3 Purpose and scope of this study Explore the utility of PSO technique and PARSE parameterization in the redesign of wind turbine blade. Obtain the Pareto-front (to be defined later) for blade section shape which can be referred to compare the performance of other airfoils. Identify the important parameters for the blade performance

23 hapter Mathematical Model. Particle swarm optimization Particle Swarm Optimization (PSO) method evolved from a simple simulation model of the movement of social groups such as birds and fishes, in which it was observed that local interactions underlie the group behavior and individual members of the group can profit from the discoveries and experiences of other members. PSO is an algorithm that uses plural points, such as genetic algorithm. For the two-objective PSO used in this research [3], each solution (particle) x is normalized between and and it is found by the following equation; x v ( t) n ( t) n x ( t) n wv v ( t) n ( t) n c r ( t) ( t) P x c r L x best best (-) The superscript t denotes the searching cycle which is similar to the generation in GA, the subscript n denotes each individual, x is the vector for design variables and v denotes the velocity vector. The terms in equation (-) is specified as follows; 3 r3 w.5 c c r, r, r 3 :uniformly distributed random numbers in the range[,] (-) PSO learns P best for exploiting the best results found so far by each of the particles and a local best L best is found for each swarm member selected from the "closest" two swarm members. The concept of closeness is calculated in terms of only one of the objective functions, while the selection of the local optima is accomplished from the two based

24 upon the other objective. The way to find L best is illustrated in Fig.-, in which we want to minimize both Objective and Objective functions. The L best for particle A (Fig. - (a)) is determined in the following procedure. First, the distance from particle A in terms of objective is compared (Fig. - (b)) and the two closest particles B and are determined. Secondly, the value of objective is compared among B and. The better one in terms of minimizing objective, is particle whose value is L best.. Objective function In order to obtain the performance of a blade section, e.g., lift and drag coefficients ( l and d ), flow field around it must be calculated. For this purpose, consumer software "IEM FD" and "FLUENT" were used. The section shape was first calculated by using the PARSE method with the parametric values resulting from PSO. Then the points at the airfoil were imported to IEM FD for grid generation. That grid was then imported to FLUENT for flow field calculation. The objective function values, and, are calculated and put back into optimization code. All components in this l d procedure will be explained in the following sections... Aerodynamics of wind turbine... Power available from wind The kinetic energy of a stream of air coming into a wind turbine with rotating surface area A T in time t can be written as (Fig.-): 3

25 where, E a AT V tv (-3) a : density of air V : wind velocity at far field Therefore, energy per unit time (Power) is; E t P air a A V T 3 (-4) The ratio between this theoretical power available in a wind stream and the power developed by the wind turbine ( P T ) is called power coefficient ( ) power power PT PT (-5) P A V air a T 3 The maximum value of power on the axial moment theory shown below [9]. is 6/7, known as the Betz limit, and can be derived based... Axial moment theory In the axial moment theory, flow field is assumed as: incompressible, steady, one-dimensional homogeneous (fluid properties are same within any cross section vertical to the flow direction) static pressure far in front and behind the rotor are considered to be equal to the atmospheric pressure frictional drag and wake behind the rotor is neglected 4

26 The letters in Fig.-3 are; V : wind velocity at section - A : area of section - P : atmospheric pressure V T : velocity at the turbine section A T : rotating surface area of the rotor P U : pressure at the upstream of the rotor P D : pressure at the downstream of the rotor V : wind velocity at section - A : area of section - From conservation mass, A a V a AT VT a AV (-6) Thrust force experienced by the rotor can be written in two forms. From the difference in the moment of incoming and outgoing winds; a AV a AT VT V V F AV (-7) a From the pressure difference at the upstream and the downstream of the rotor, U D A T F P P (-8) By applying Bernoulli s equation, av avt P PU (-9) 5

27 av avt P PD (-) From (-9) and (-), P U P D a V V Substitute (-) in (-8), a V F V A T (-) (-) From (-7) and (-) we get, V T V V (-3) Introduce the axial induction factor a. a V V V T (-4) This parameter indicates the degree with which the wind velocity at the upstream of the rotor is slowed down by the turbine. From (-4), V T V a (-5) From (-3) and (-5), a V V ( ) V V T V V a (-6) Power developed by the turbine due to the transfer of kinetic energy is; P Substitute P T T a AT VT V V (-7) V T and V from (-5) and (-6), A V 3 4a a a AT V av V a a T (-8) 6

28 From (-5), P T 3 a ATV power (-9) By comparing (-8) and (-9), power 4a a (-) For power to be maximum, d d power and power da da (-) d da a d 3 4 a a 4 a 8a 4a a 6a 4 43a a, 3 da d da power 8 ( at a ) 4a ( at a ) power is maximum when a. 3 Assign a to (-). 3 6 power _ max % (-) P 3 T _ max a AT V (-3) 7 As mentioned by Theodorsen [8], axial momentum theory cannot be considered as a mathematical limit for the description of a wind turbine. However, this theory has an advantage that it can give the general properties of the system without conducting a heavy FD calculation for the entire flow field. 7

29 ...3 Power created by a blade element In this section, the effects of rotor shape will be explained by the blade element theory[9]. In the blade element theory, the rotor is split in the direction of radius, each of length dr, and the torque and power created by these elements are the differential rotor torque and differential rotor power. The assumptions made in the blade element theory are: There is no interaction between the analyses of neighboring elements (Quasi-D approximation). The forces exerted on the blade elements by the flow stream are determined solely by the two-dimensional lift and drag characteristics of the blade element airfoil shape and orientation relative to the incoming flow. Let us consider a situation that a uniform wind of velocity V T is perpendicular to the plane of wind turbine that is rotating at [rad/s] (Fig.-4). The relative flow velocity with respect to airfoil, V rel that each blade element experiences are a combination of VT and the rotating velocity r (Fig.-5). Letters in Fig.-5 refers to: By definition, : angle between Vrel and r l : lift coefficient of the blade element d : drag coefficient of the blade element : angle of attack (AOA) of the blade element V rel r T V (-4) 8

30 V T tan (-5) r Differential torque dq at r in the direction of can be expressed as; rel dq B a V sin cos r dr l d (-6) where, B : number of blades : chord length Since the differential power P T dris; dr dq P T (-7) From (-6) and (-7), P T a rel dr B V sin cos r dr l d (-8) From (-8), it can be seen that high sin and low cos will provide higher l d power for the same chord length. As seen in Fig.-5, is a function of V T, and r. Therefore, sin and cos only depends on the operating condition and not on the airfoil. As a conclusion, airfoil with high l and low d is preferable....4 Optimization condition for wind turbine airfoil In this section, we will unite the knowledge from the previous two sections and discuss the desired characteristics of wind turbine airfoil. First, we will differentiate (- 9) for r. At r, the differential rotating surface area is: r dr r r dr dr rdr da( r) (-9) 9

31 Therefore, P T (-3) 3 dr a rdrv power From (-8) and (-3), V power (-3) BV sin cos rel Introduce tip speed ratio which is, l 3 d R (-3) V Where R is the overall length (center of rotation to tip) of the rotor. From (-5), (-4) and (-3), V rel V V r a r V a R R (-33) Substitute (-33) in (-3), Power R (-34) r B a l sin d cos R For example, Fig.-6 shows how the chord length would change along the radius direction for the following condition: power R 5m B 3 5 a (-35) 3

32 It should be noted that in Fig.-6, sin l d cos l sin for simplification. To obtain a constant l value along the radius direction, the blade must be twisted so that the angle of attack for each blade section will be the same. Flow field around each airfoil section can be characterized by Reynolds number. a V Re rel (-36) Where is the dynamic viscosity of the air. From (-33), (-34) and (-36), Re a V l a av powerr B sin cos d r R B r R power a sin cos R l d (-37) Equation (-37) is independent of r. This means that as long as chord length follows equation (-34), the same optimization condition can be used at any r-position along the turbine blade. In sum, under the assumptions we have made, the airfoil for wind turbine must have the following characteristics; High l Low d And the same Reynolds number and angle of attack can be used from the hub to the tip, since the sectional Reynolds number is independent of r. 3

33 .. PARSE airfoil... The original PARSE airfoil The PARSE method [] defines the shape of an airfoil by eleven parameters shown in Fig. -7. By these eleven parameters, the airfoil shape is given by the following equation, upper and lower independently. 6 n/ Z parsec an( p) x (-38) n where the coefficients a n is determined by the given geometrical parameters shown in Fig. -7. Equation (-38) can be solved in matrix form as follows; a and ( P) r le (-39) x 3 up 3 x 3 xup 4 3 up x 5 5 up x 5 x up up x 7 7 up x 35 x up 3 up x 9 9 up x 63 x up 5 up x up x 99 x 4 up 9 7 up zup rle x up r le a p ( ) z TE zte rle a p 3( ) a p rle / x up 4( ) a p 3 5( ) z xxup rle / x up a ( p) 4 6 tan( TE TE ) rle Equation (-4) is solved by Gauss elimination. (-4)... Modified PARSE airfoil Since the PARSE method was not especially designed to parameterize airfoils for wind turbine, we noticed that this method must be modified to increase its robustness for our purpose. As seen in Fig.-7, the center of the leading edge radius is always on the x-axis for the original PARSE method. For our optimization, the leading edge radius 3

34 was defined by two parameters ( r _, le up r le _ low ) instead of one parameter ( r le ) (Fig. -8). Table.- shows the performance of three different airfoils. ase A is a shape occurred during the optimization which has a different value for r _ and le up r le _ low. ase B has the same value for r _ and le up r le _ which is the low le up r _ value of case A. Likewise, case has the same value for r _ and le up r _ which is the le low r _ value of case A. The shapes of le low these three airfoils that have the same PARSE parameters except r _ and le up r _, are le low shown in Fig.-9. Figure - compares the l and d values obtained from these three airfoils (the details of calculation condition will be explained in chapter 3). From Fig.- it can be seen that case A, the shape described by the modified PARSE method, is not dominated by either case B or case. The modified PARSE method can extend the robustness of the PARSE method....3 Thickness calculation for PARSE airfoil Maximum thickness is often a constraint in practical wind turbine designs due to structural considerations []. Through the optimization, it was noticed that the maximum thickness of an airfoil has a strong effect on its aerodynamic performance. In the PARSE method, the upper surface and lower surface are defined independently. Therefore, the thickness of the airfoil must be calculated from its shape. Throughout this paper, the term thickness and camber line are used according to the following definition: Thickness: Distance between two points on airfoil that intersects with a vertical line from the camber line. 33

35 amber line: Line that connects the center of circle inscribed in the airfoil (See Fig. -) The procedure for calculating the thickness is as follows;. The airfoil is equally split at 5 locations along the chord line (Fig.-a).. For each of the five locations, say location A, a line is drawn perpendicularly from the lower surface (Fig.-b) 3. Another line is perpendicularly drawn from the upper surface at an arbitral point B and the intersection with the previous line will be called D (Fig.-c). 4. The distance from A to D, B to D will be called r and r respectively (Fig.- d). 5. Finally, find the Point B that gives r r by Golden section search []. Thickness T at this location will be r (equal to r ) (Fig.-e). 6. By using the thickness at all five locations and x= (which gives T =zero), the thickness can be given by the following equation. 6 n T ( x) b ( ) / n T x (-4) n where the coefficients b n is determined by the thickness T at different x locations. Equation (-4) can be solved in matrix form as follows; 34

36 x T x T x T x T x T b b b b b b (-4) Equation (-4) is solved by Gauss elimination...3 Flow field calculation As mentioned in Section.., flow field around the airfoil was calculated by the RNG k-εmodel [] using FLUENT. This turbulence model uses wall function near the wall which reduces the computational time significantly. First, the governing equations for the RNG k-εmodel will be explained and followed by the explanation of near wall treatment. Finally, the grid used in the flow field calculation that satisfies the requirement of the RNG k-εmodel will be explained...3. RNG k-εmodel The assumptions made in this calculation are; incompressible ( 3 /.5 m kg ) steady

37 36 constant viscosity ( s m kg / ) Under these conditions, the governing equations are continuity equation, y v x u (-43) momentum equation, y u x u x P y u v x u u (-44) y v x v y P y v v x v u (-45) The transport equations for turbulence kinetic energy k and its dissipation rate are, 4 x v y u y k y x k x kv y ku x t k t k t (-46) k x v y u k y y x x v y u x t t t * 4 (-47) where,

38 37 x v y u k k t k * (-48) Only is derived experimentally and every other constant coefficients in RNG k-ε model are derived analytically [3]...3. Near wall treatment [4] For computing points near the wall, the standard wall function option (which is based on the proposal of Launder and Spalding [5]) of FLUENT was used. For momentum equation, the law-of-the-wall for mean velocity yields: * * ln Ey U (-49) where, / 4 * w P k p U U (-5) P k p y y 4 * (-5)

39 and = von Kármán constant (=.487) E = empirical constant (=9.793) U P = mean velocity of the fluid at point P k P = turbulence kinetic energy at point P y P = distance from point P to the wall = dynamic viscosity of the fluid where subscript P denotes the properties at wall adjacent cell. The logarithmic law of mean velocity is known to be valid for; 3 y * 3 (-5) Boundary condition at wall for k and transport equation are: k n (-53) where n is the local coordinate normal to the wall. The production of k, which is the third term in the right of equation (-46), is computed from; t u v 4 y x U w y w w 4 k p y P (-54) and is computed from; 3 4 P 3 p k P (-55) y 38

40 ..3.3 Discretization Discretization methods for each component are: Pressure: second order center differentiation Momentum, k and transport equation: second order upwind differentiation..3.4 Grid generation When generating a mesh around an airfoil, the following components must be decided; Type of grid (structured grid, unstructured grid, O-grid, -grid, etc.) Domain size Boundary conditions The number and distribution of points around the airfoil The number and distribution of points throughout the domain. And the grid should give the following results under the RNG k - method. The solution should be grid-independent The solution should give a good match with available experimental data. If the experimental data was not available, a comparison with other computed solution of a similar problem should be pursued. The solution must satisfy the criteria for wall function (-5) In order to have a good control of distance from the wall to its adjacent cell, structure grid should be used if possible for the RNG k - model. A -type grid (Fig.-3 and Fig.- 4) was used for this research since the performance of airfoil highly depends on its 39

41 leading edge resolution (-type grid allows the user to have many points near the leading edge unlike H-type grid) and the PARSE airfoil has a sharp trailing edge (O-type grid is not good for sharp trailing edge airfoils). Fig.-3 explains the domain size. The chord length of the airfoil is set to. and it exists in the region between X=. and.. The number of points along the i-direction was 46 points, of which 3 points were put around the airfoil. For j-direction, 7 points were used. Fig.-4 shows the grid near the airfoil. The total number of grids used in the flow calculation is 3,. There are three types of boundary condition in this calculation as shown in Fig.-5. At the velocity inlet boundary, magnitude and direction of the flow is specified. At the pressure outlet boundary, pressure is set at atm and the magnitude and direction of the flow are free. At the wall boundary, the wall function (section..3.) is introduced. The validation process of this grid is explained in section Wind turbine airfoil shape optimization Fig.-6 shows the flow chart of wind turbine airfoil shape optimization. Each component will be explained in the following sections..3. Flow condition for optimization The optimization was done at; Re. 5.3 deg 6 (-56) 4

42 where is the angle of attack. This is the Re at which the wind turbine airfoil S89 was designed [5] and the angle of attack is when the S89 provides the highest value of lift to drag ratio L/D (which is often used for installation on actual wind turbine)..3. Geometrical parameters for optimization The airfoil shape was defined by the modified PARSE method (see...). Out of the parameters in the modified PARSE method (Fig.-8), fixed at zero for the following reasons; zte and z TE were Although there are some research conducted for the blunt trailing edge wind turbine air foils [6], the accuracy of RNG k-εmethod is questionable when the airfoil has severe separation [8]. The merit in having a blunt trailing edge is mostly on structural reasons, which is not computed in our current optimization. The reason of having z TE at zero is because otherwise the chord length will change from to a larger value. In sum, the airfoil was defined by PARSE parameters and the bounds for each of these parameters are; 4

43 . r. Z. Z.7 X.5..7 X.3 Z.5 Z. r le _ up TE xxup up up TE le _ lo lo lo xxlo (-57).3.3 Objective functions for optimization As explained in chapter...3, we want a high l and low d airfoil. To make the optimization problem into a minimization problem is customary in the field of optimization (and it will make figures easier to understand). Therefore, the objective functions become l and d..3.4 oncept of domination Since we want to minimize both l and d, the problem we want to solve is multi-objective problem (more than one objective function). Most multi-objective optimizations use the concept of domination, which is defined as follows [7]: A solution x true: is said to dominate the other solution x, if both condition and are. The solution. The solution x is no worse than x is strictly better than For example, see Fig.-7. It can be said that; x in all objectives. x in at least one objective 4

44 Solution D is dominated by solution B Solution E is dominated by solution Solutions that cannot be dominated by any other solution are called Pareto-optimal solutions and the curve that connects these solutions are called Pareto-optimal front. Solving a multi-objective problem is to find the Pareto-optimal front and to find out which set of parameters will give those solutions..3.5 Accuracy check criteria for optimization During the optimization process, some shapes that are not feasible must be taken out from the optimization pool. These can be categorized in three groups: When the airfoil shape is unphysical When the objective function could not be computed accurately When the maximum thickness of an airfoil exceeded its limit. When these shapes occur, a penalty function is applied so that they will be considered as bad solution Unphysical airfoil Since the PARSE method describes the shape of an airfoil by the upper curve and lower curve independently, the two curves may intersect (Fig.-8). Obviously, such an airfoil is unphysical. In this case, the objective function will be given as: l d.. (-58) 43

45 Since these objective function values are extremely bad, the particle will tend to stay away from such solution. This criterion was checked by the following procedure. First, the upper curve and the lower curve were split by points, which is equally spaced along the cord line. Second, at each of these locations, it was checked weather the height of the upper curve point is bigger than the height of the corresponding point on the lower surface or not Airfoil that does not give an accurate solution Flow field is calculated for iterations. Fig.-9 and Fig.- shows how the force in the direction of chord line F x and the force vertical to the chord line F y value evolve as the number of iteration increases. In the case of Fig.-9 and Fig.- it appears that we can trust the value obtained at iterations. On the other hand, in the case of Fig.-, the F y value is still oscillating and the value at iterations cannot be used as an objective function. To make this judgment automatically during the optimization, the following criteria were used. F F x _ F y _ F F x _ F y _ x _ 95 y _ 95.. (-59) Where the subscripts 95 and on F x and Fy means the values at 95 iterations and iterations respectively. 44

46 Another criterion that must be checked is (-5), whether the flow field satisfies the applicability of the wall function or not. If the solution fails to satisfy (-5), the solution is not reliable whether it has converged or not. When the solution fails to satisfy (-5) or (-59), the following penalty function is applied: l d d l 5.. (-6) This means that solution will be considered bad (but not as bad as (-58)), and the particles will move towards a more accurate solution Maximum thickness of airfoil exceeds its limit During this research, we noticed that the maximum thickness of an airfoil has a strong effect on its performance. When we want a solution that has a maximum thickness of exceeding %, the airfoils that have a maximum thickness less than % should be taken out of the population. This is done by giving the following penalty function: l d.. (-6).3.6 Parallel computing by MPI (Message Passing Interface) As mentioned in section.., the capability of utilizing parallel computing effectively is one of the major advantages of PSO. In this research, grid generation and flow field calculation were performed using 4 processor elements (PE). 45

47 Let us consider a situation when there are a total number of N particles and they are numbered by integers, through N. First, the particles are divided into 4 groups (say nid, nid, nid3 and nid 4 ): N nid: through 4 N nid: through 4 N nid3: through 4 3N nid4: through 4 N 4 3N 4 N (-6) For each of these group members, the following process is parallelized. create the airfoil shape from the location of PSO particles by the modified PARSE method grid generation (IEM FD) flow field calculation (FLUENT) compute the function values ( l and d ) Once the function values are all computed for every group, the results are combined in one file and put back into the optimization. Since grid generation and flow field calculation is where most of the computing time is used, parallelizing this process greatly reduces the computing time. 46

48 Table - l and d comparison for different r le _ up and le low r _. r le_up r le_low l d case A case B case

49 Figure - How particle finds its local best (for particle A) 48

50 Figure - Air coming into a wind turbine 49

51 V A P PU V T A T PD V A P Figure -3 Axial stream tube model 5

52 Figure -4 Air flow vertically coming into the rotating surface 5

53 Figure -5 Relevant velocity for blade section 5

54 4 l =.8 l =. l =. radius [m] 4 hord length [m] Figure -6 hord length distribution along the radius direction 53

55 Figure -7 PARSE airfoil geometry defined by basic parameters: leading edge radius, upper and lower crest location including curvature there, trailing edge coordinate (at X = ), thickness, direction and wedge angle. [] 54

56 Figure -8 Modified PARSE airfoil geometry defined by basic parameters: compared to Figure -7, r le has been split into r le _ up and r le _ low. ([] modified ) 55

57 .5 case A case B case z.5.5 x Figure -9 Shape of the three airfoils (case A, B and ) 56

58 .85 case A case B case.84 d l Figure - Objective function values of the three airfoils (case A, B and ) 57

59 .5 amber line Z.5.5 X Figure - Sketch of camber line 58

60 Figure - How thickness is calculated for PARSE airfoil 59

61 Figure -3 Domain size 6

62 Figure -4 Grid near the airfoil 6

63 Figure -5 Boundary conditions 6

64 START Define flow condition Define geometrical parameters (modfied PARSE) Grid generation (icem cfd) Flow field calculation (FLUENT) Post procesing (compute the objective functions) Accuracy check PSO optimizer END Figure -6 Flowchart for wind turbine airfoil shape optimization 63

65 6 A 4 D f (minimize) B E 4 6 f (minimize) Figure -7 oncept of domination 64

66 .5 z.5.5 x Figure -8 Example of unphysical airfoil shape 65

67 F x F x..4 number of iteration Figure -9 Example of how F x value converges 66

68 F y.6 F y.4 number of iteration Figure - Example of how F y value converges 67

69 F y F y number of iteration Figure - Example of unconverged solution 68

70 hapter 3 omputed results 3. Validation of components 3.. A test of two objective PSO by Hu and Eberhart Two-objective PSO by Hu and Eberhart [3] was tested against a cantilever design problem mentioned in Deb s [7] work. Fig.3- shows a sketch of this problem. A cylindrical shaped cantilever of length l and diameter d is under force P at the end. The objective is to minimize the weight of the cylinder and deflection at the end by changing the parameters d and l. In other words: Objective functions d minimize: f( d, l) l 4 64Pl minimize: f ( d, l) 3Ed 3 4 (3-) Range of parameters mm d 5 mm mm l mm 3Pl max S 3 y Subject to: d max 78 kg / m P. kn onstant parameters: E 7 GPa S y 3 MPa max 5 mm 3 (3-) (3-3) (3-4) where, : density E : Young's modulus S y : allowable strength : deflection : deflection limit max 69

71 Some objective function values that satisfy (3-) may not satisfy (3-3). When the objective function values satisfy all of the constraints, they are called feasible solutions. For instance, when d mm and l 4mm : 3Pl 3 max 3 d. 3 64Pl Ed MPa S mm y max (3-5) Since (3-5) does not satisfy (3-3), d mm and l 4mm is an unfeasible solution. On the other hand, when d 3mm and l 3mm : 3Pl 3 max 3 d Pl Ed MPa S mm y max (3-6) Because this solution satisfies equation (3-3), d 3mm and l 3mm, it is a feasible solution. The "Directly computed feasible solution" in Fig.3- was obtained by the following procedure: ) Define dthrough d and l through l the range defined in (3-). so that they are uniformly distributed in d., d.4, d3.8,..., d l., l.8, l3.6,..., l.5 m. m (3-7) ) ompute the function values for all combinations between d through d and l through l. 3) The function values that satisfy the constraints given by (3-3) is treated as a directly computed feasible solution. 7

72 When solving a complicated problem, it is not realistic to compute all of the feasible solutions because it requires too long of a computational time. Optimization techniques are used so that it can find the Pareto-front without computing all the feasible solutions. Only ten particles are used for this cantilever problem to make sure particles are moving towards the Pareto-front. Fig.3- shows how the particles move along the increasing number of iterations. In the beginning, particles are randomly distributed and as the number of iteration increases, they become closer to the Pareto-front. After iterations, every feasible particle lies near the Pareto front. Fig. 3-3 shows a closer look of the particles at iterations. By solving this problem, it was confirmed that the scheme introduced works well. 3.. Flow field calculation around S89 air foil To validate the flow filed calculation process, we referred to a paper written by Wolfe et al [8]. In this paper, a D wind tunnel experiment of the S89 airfoil was compared with the inviscid Euler and the viscous laminar, turbulent and laminarturbulent mixed conditions. First we ran an Euler calculation in order to make sure we are able to describe the shape of the S89 airfoil accurately. Then, we ran the RNG-k-ε calculation and compared the result with experimental data Method of representing S89 airfoil Airfoil coordinates of the S89 was originally given by Somers [5] by 6 points (Table 3-). When creating a mesh around this airfoil, the points in between the given coordinates should be interpolated. Fig.3-4 and Fig.3-5 compares the airfoil shape 7

73 described by GAMBIT and IEM FD. Although the curves described by both software go through the official coordinates of the S89, there are some differences near the leading edge (Fig.3-5). By comparing the curve given by both software, interpolation by GAMBIT was smoother than that of IEMFD. The shape of the S89 airfoil was redefined by points as shown in Table 3- by the interpolation function of GAMBIT. To make sure this interpolation is correct. An Euler calculation was done so that we can compare the P distribution given by Wolfe et.al. [8] 3... Flow field calculation around S89 air foil (Euler) Pressure coefficient P distribution around the S89 airfoil was computed by solving Euler s equation for different angle of attacks. Since the experimental result by Somers [5] was obtained at 6 Re., this Reynolds number was used to define the inlet velocity for Euler calculation. For air at 5, atm:.8 m U Re 5.5kg / m Pa s 3. 6 (3-8) U 9.7m / s.5 (3-9) Where, : viscosity : density : chord length U: inlet velocity 7

74 Fig. 3-6 through Fig. 3-8 compares the P distribution obtained by Euler calculation and the experimental result by Somers [4]. For 5.3 deg, the Euler calculation by Wolfe and Ochs[8] is also included for comparison. Although experimental data of P distribution was given at 9. deg, 4.4 deg, and.5 deg as well, the Euler calculation failed to converge at 9. deg (Fig.3-9). This is because the inviscid calculation does not provide separation that occurs in the experiment. It can be said from Fig. 3-6 through Fig. 3-8 that the computed P distribution matches experimental results. Most importantly, the P distribution given by our Euler calculation matched very well with the P distribution given by Wolfe and Ochs [8] using Euler calculation. This means that the official coordinates of the S89 were interpolated in a same manner Flow field calculation around S89 air foil (RNG k-ε) Flow field around the S89 airfoil was calculated with the same grid-generator and fluid solver with the optimization case. The results are compared with experimental results [5] at Re. 6. omparisons were done in ) P distribution, ) l at different angles of attack, 3) Fig.3- through Fig.3-4: Fig.3-5: Fig.3-6: d at different angles of attack. P distribution at angle of attack of,., 5.3, 9. and 4.4 deg. l at different angles of attack d at different angles of attack k-ε seems good for all the angles of attacks computed as far as concerned. For, P l there is some discrepancy when the angle of attack is close to 73 5 where flow may be

75 separated at certain portion of the airfoil. The use of wall function is not justified at those locations. Fig.3-6 also shows k-ε over estimated d in the range of. It should be noted that, the experiment was done in a wind tunnel and the flow involved laminarturbulent transition where RNG k-εmodel is assuming a fully turbulent flow along the entire airfoil surface. This difference is causing a higher estimation of d value (Fig.3-6) around 5.3 at which optimization was performed. Note that the objective of this research is to find the nondominated set of airfoils i.e. which airfoils are better than the others. Within the sets of airfoils computed, the properties between the laminar to the turbulent regions are unlikely to vary drastically at the same Re. We feel that RNG k-ε model is capable for this optimization exercise despite some discrepancy on the absolute magnitude of d computed Grid sensitivity To make sure that the flow field calculation is grid independent, five different grids were tested. The "base grid" are the grid described in section Details of the other 4 grids and computed results are compared. These comparisons are described next OL grid (Outlet Length times) In the "OL grid", the length from the trailing edge to the outlet boundary is times longer than the "base grid"(fig.3-7). Fig.3-8 and Fig.3-9 compares the l and d value against angle of attack. It can be seen from Fig.3-8 and Fig.3-9 that the solution obtained from "OL grid" is identical to the solution obtained by the "base 74

76 grid". In other words, the "base grid" has a sufficient length from the trailing edge to the outlet boundary IL grid (Inlet Length times) In "IL grid", the length from the trailing edge to the inlet boundary is times longer than the "base grid" and the number of points vertical from the airfoil is increased by times as well (Fig.3-). Fig.3- and Fig.3- compares the l and d value against the angle of attack. It can be seen from Fig.3- and Fig.3- that the solution obtained from "IL grid" is identical to the solution obtained by the "base grid". In other words, the "base grid" has a sufficient length from the trailing edge to the inlet boundary D grid (Grid density times) In the "D grid", the number of points vertical to the airfoil is increased by two times (Fig.3-3). Fig.3-4 and Fig.3-5 compare the l and d value against the angle of attack. It can be seen from Fig.3-4 and Fig.3-5 that the solution obtained from the "D grid" is identical to the solution obtained by the "base grid". In another words, the "base grid" has a sufficient number of points vertical to the airfoil B grid (different Boundary ondition) In the "B grid", the boundary condition is changed as shown in Fig.3-6 and everything besides the boundary condition is the same as the "base grid". Fig.3-7 and Fig.3-8 compares the l and d value against the angle of attack. At the angle of attack 75

77 deg, the B grid did not provide a stable solution. Although the l and d values given by the B grid are similar to that of the base grid; to maintain the capability of getting the solution at deg, we would prefer to use the "base grid". 3. Wind turbine airfoil shape optimization 6 Wind turbine airfoil shape was optimized at Re. with an angle of attack at 5.3 deg. Two sets of optimization were conducted based on the thickness constrain. The airfoil shapes and its performance were compared with that of the S89 airfoil, which is.9% thick ( according to the thickness defined in section...3). 3.. % thick condition In % thick condition, the maximum thickness of airfoils was restricted by; % maximum thickness 5% (3-) where % means % of the chord length. particles were used for 37 generation. Fig.3-9 shows all of the computed function values and Fig.3-3 shows all of the non-dominated solutions. The non-dominated solutions obtained by the optimization are numbered in Fig.3-3. It can be seen from Fig.3-3 that the performance of the S89 is very close to the line created by the non-dominated solutions. Table 3-3 shows the thickness of nondominated solutions. It can be seen that except the solutions -7 and -, the thickness is between % and %. Fig.3-3 through Fig.3-4 shows the shape of nondominated solutions and its performance for different angle of attack. The non-dominated solutions can be divided into three groups; 76

78 low d and low l airfoil (-, -) Although these two shapes have low d value compared to the S89 at the optimization condition (5.3 deg), the of the S89. For airfoil is undesirable. d value at the lower angle of attack exceeds that l, it is always lower than the S89. For the wind turbine, this group of high l and high d airfoil (-9, -) The way this group of airfoils produce torque is close to that of drag based wind turbines (ex. paddle type). Although this could be an interesting field of study, it conflicts with the fact that we are parameterizing the blade shape for lift based wind turbines. The performance of airfoils in this category is considered poor. Airfoil -9 also has a shape that is not good from the structure point of view. reasonable l and d value (-3 through -8) The last category of airfoils is the type that does not fit either of the categories above. Among the 6 airfoils in this category, the performance of -4 is very close to that of the S89. If we followed the approximated equation of (-7), -8 will generate the highest torque. 3.. % thick condition In the % thick condition, the maximum thickness of airfoils was restricted as: % maximum thickness 5% (3-) 77

79 where, % means % of the chord length. particles were used for 54 generations. Fig.3-4 shows all of the computed function values and Fig.3-43 shows all of the non-dominated solutions. From Fig.3-43 it can be seen that in this condition, there are many solutions that dominate the function value of the S89. Fig.3-44 shows non-dominated solutions that dominate the S89 s function value. They are named in Fig Fig.3-46 through Fig.3-55 show the shapes and the corresponding performances for different angles of attack of the solutions in Fig The non-dominated solutions that dominate the S89 share the same characteristic, in that they are about half as thick as the S89. The shape - has about the same level of d as S89 although its l performance is constantly. 3 to. 4 higher. The -8 and -9 airfoils might have some structural problem around x=.5 since they are thin around this region, indicating that structural calculation is needed to give a conclusion in this aspect omparison of % and % thick condition Fig.3-56 compares the non-dominated solutions obtained by the % thick condition and the % thick condition. The shape of the curve made by nondominated solutions is similar, and the % thick case is lower in d and higher in compared to the % thick case. It can be seen vividly that there is a strong correlation in the performance of the airfoil and its maximum thickness. By calculating a required thickness from a structural point of view and choosing the wind turbine section shape l 78

80 accordingly, the overall performance of a wind turbine shall increase compared to just using a single airfoil shape for the whole turbine. 79

81 Table 3- The original S89 airfoil coordinates by Somers Upper surface Lower surface x/c z/c x/c z/c

82 Table 3- Interpolated S89 airfoil coordinates using GAMBIT Upper Surface Upper Surface (continued) Lower Surface Lower Surface (continued) x/c z/c x/c z/c x/c z/c x/c z/c E E E

83 Table 3-3 Thickness of nondominated solutions in "%thick condition" name thickness %

84 P l Fig. 3- Sketch of the cantilever design problem d 83

85 .4 f. directly computed feasible solution ITER= ITER= ITER= f Fig. 3- Particle movement along the increasing number of iteration 84

86 . directly computed feasible solution ITER= f f Fig. 3-3 loser look of the Pareto front at iteration 85

87 .5 S89 original coordinate IEM FD GAMBIT Z.5 Fig. 3-4 omparison of interpolation by IEM FD and GAMBIT (far view) X 86

88 .4 S89 original coordinate IEM FD GAMBIT Z...4 X Fig. 3-5 omparison of interpolation by IEM FD and GAMBIT (close view) 87

89 Euler calculation Experimental [5] p X Fig. 3-6 P distribution by Euler calculation (angle of attack of [deg]) 88

90 Euler calculation Experimental [5] p Fig. 3-7 X P distribution by Euler calculation (angle of attack of. [deg]) 89

91 Euler calculation Experimental [5] Euler calculation by Wolfe et. al. [9] p Fig. 3-8 X P distribution by Euler calculation (angle of attack of 5.3 [deg]) 9

92 F Z.5 F Z.5 number of iteration Fig. 3-9 How F Z does not converge at angle of attack of 9. [deg] F is the non-dimensional force in the z-direction of Fig.3-4) ( Z 9

93 RNG k calculation Experimental [5] p Fig. 3- X P distribution by RNG k-ε calculation (angle of attack of [deg]) 9

94 RNG k calculation Experimental [5] p Fig. 3- X P distribution by RNG k-ε calculation (angle of attack of. [deg]) 93

95 RNG k calculation Experimental [5] p X Fig. 3- P distribution by RNG k-ε calculation (angle of attack of 5.3 [deg]) 94

96 p RNG k calculation Experimental [5] X Fig. 3-3 P distribution by RNG k-ε calculation (angle of attack of 9. [deg]) 95

97 p 4 6 RNG k calculation Experimental [5] X Fig. 3-4 P distribution by RNG k-ε calculation (angle of attack of 4.4 [deg]) 96

98 l RNG k calculation Experimental [5] 5 5 angle of attack [deg] Fig. 3-5 l comparison between RNG k-epsilon calculation and experiment 97

99 .8 RNG k calculation Experimental [5] d.4 Fig angle of attack [deg] d comparison between RNG k-epsilon calculation and experiment 98

100 Fig. 3-7 Grid comparison between "OL" grid and "base grid" 99

101 base grid OL grid l angle of attack [deg] Fig. 3-8 l comparison between "OL grid" and "base grid"

102 .8 base grid OL grid.6 d.4. angle of attack [deg] Fig. 3-9 d comparison between "OL grid" and "base grid"

103 Fig. 3- Grid comparison between "IL" grid and "base grid"

104 base grid IL grid l angle of attack [deg] Fig. 3- l comparison between "IL grid" and "base grid" 3

105 .8 base grid IL grid.6 d.4. angle of attack [deg] Fig. 3- d comparison between "IL grid" and "base grid" 4

106 Fig. 3-3 Grid comparison between "D" grid and "base grid" 5